The Nearby Lagrangian Conjecture: A Start

The nearby Lagrangian conjecture asks whether every closed exact Lagrangian in the cotangent bundle of a closed manifold is Hamiltonian isotopic to the zero section. Although relatively few results on this conjecture are known, some progress has been made over the years. In files recently released by OpenAI, a counterexample to the conjecture is proposed using the machinery of tube-type generating functions and the K-theory of spectra (with Waldhausen’s stable tube space providing a model), based on a sequence of earlier works. In this talk, I will survey the history and previous discussions on this topic and discuss how the proposed counterexample works. Then I will try to explain why this is simply the start of the entire field.